Arduino Esplora

Can I connect more then 20 LED lights on Arduino Esplora?

It doesn't really have general purpose I/O pins, so probably not a good option if you want something 20 general purpose I/O pins.

I would look at Adafruit Neopixels. You can connect as many as you want given Esplora's memory and you must supply enough power via an external source. You could use one or both digital pins on the Tinkerkit outputs or digital pins on the "Display Header". See http://learn.adafruit.com/adafruit-neopixel-uberguide and http://21stdigitalhome.blogspot.com/ (search for Esplora for documenting the pins you might use).